Abstract

The boundedness of the various operators on -Morrey spaces is considered in the framework of the Littlewood-Paley decompositions. First, the Littlewood-Paley characterization of -Morrey-Campanato spaces is established. As an application, the boundedness of Riesz potential operators is revisted. Also, a characterization of -Lipschitz spaces is obtained: and, as an application, the boundedness of Riesz potential operators on -Lipschitz spaces is discussed.

1. Introduction

One of the main aims of the present paper is to investigate the structure of the -spaces by proving Theorem 1 below. To state the main result, we need the following setup. We write
for and . We abbreviate to . For a set , we denote by its indicator function. The space denotes the set of all compactly supported -functions supported on . Now we formulate our main result in the simplest form.

Theorem 1. Let , and . Let be a function such that
If a measurable function belongs to , that is, satisfies
then and
where denotes the inverse Fourier transform and the implicit constant in does not depend on .

The space appearing in Theorem 1 is a special case of spaces. Theory of spaces stems from the Beurling work [3]. Beurling introduced the space together with its predual so-called the Beurling algebra [3]. Later, to extend Wiener’s ideas [4, 5] which describe the behavior of functions at infinity, Feichtinger [6] gave an equivalent norm on , which is a special case of norms to describe nonhomogeneous Herz spaces introduced in [7]. On the other hand, Lu and Yang [8, 9] introduced the central bounded mean oscillation space with the norm
Recently, in [10], -Morrey-Campanato spaces have been introduced to unify central Morrey spaces, central bounded mean oscillation spaces, and usual Morrey-Campanato spaces. Using -spaces, we can study both local and global regularities of functions simultaneously. For example, when we consider , the underlying norm given by
measures local regularities of functions, and the parameter plays the role of global regularity. Moreover, the proof of Theorem 13 reveals this aspect of our studying both local and global regularities of functions simultaneously.

Theorem 1 concerns the Littlewood-Paley theory. The Littlewood-Paley theory is one of the most powerful tools in harmonic analysis. Roughly speaking, this is a technique of transforming functions into good ones in order to measure the norms. Another side of Littlewood-Paley theory is that the functions are broken into good pieces of functions. We shall illustrate that the Littlewood-Paley theory is very useful by applying this to the fractional integral operator of order . The fractional integral operator , which is given by
plays an important role not only in harmonic analysis but also in partial differential equations. It is well known that can be seen as the inverse operator of modulo a multiplicative constant, and hence has the smoothing effect. However, due to this smoothing effect, still seems to have a lot to be investigated.

To describe the related function spaces and formulate Theorem 1 in the full statement, we now fix some more notations. Since cubes play the central role, let us fix our notation of cubes. We denote by the set of all cubes of the form with and . Also, we denote by the set of dyadic cubes:
where to define the right-hand side we used the Minkowski sum. Given , we denote by the center of a cube and by its volume. The notation stands for . Given a set , we define

With these notations in mind, let us recall the definition of the function space and related function spaces defined in [10]. Here we redefine in terms of dyadic cubes. However, a geometric observation shows that the definition of through dyadic cubes and that through cubes are equivalent.

Definition 2. Let , , and . Define as the set of all such that , where
Define in analogy with . The norm is given by

As we remarked above, (3) and (10) are equivalent definitions, and the same can be said for weak type spaces. Namely,
for any measurable function on . In view of (12), we identify the right-hand side and the left-hand side in these formulas. Note that from the definition of norms, we have
for any measurable function on .

We need to pay attention to the word “local,” when we discuss . Burenkov and Guliyev, together with their successors, investigated local Morrey-type spaces in [11]. By “local” in [11], they meant that they defined the “local Morrey-type” norm by
which indicates that appearing in the definition is restricted to all balls centered at the origin. However, by “local” in the present paper we meant that we measure the regularity of functions by (6). Nowadays there are many different definitions related to classical Morrey spaces, so we need to carefully distinguish the different definitions and the names given to the definitions. See [12] for related usage of the word “local” such as central mean oscillation, central , -central bounded mean oscillation, and -central Morrey spaces.

The goal of the present paper is to show that these function spaces fall under the scope of the Littlewood-Paley theory. As an application of this fact we show the boundedness property of and singular integral operators. The Littlewood-Paley theory is a powerful tool to investigate the boundedness property of . To consider the connection between and the Littlewood-Paley theory, we present definitions. Here and below we use the definition of the Fourier transform below for definiteness:
Let be a function such that
Following [13], we define
It may be helpful to observe that
for each .

In the present paper, the following function space of Littlewood-Paley type will play a key role. Here and below we denote for a cube . Observe that

Definition 3. Let . Given , set
The function space denotes the set of all tempered distributions for which the quantity is finite.

Note that Definition 3 is closely related to the space defined by Yang and Yuan and the function space is investigated in [14–21].

In Theorem 1, we did not mention what happens if , and the right-hand side of (4) is finite. Here including this problem, we reformulate and reinforce Theorem 1.

Theorem 4. Let . Then the spaces and are isomorphic. More precisely, we have the following.(a) in the sense of continuous embedding. (b) in the sense of continuous embedding. (c)Let . If , then . Conversely, if , then there exists a polynomial such that , and in this case the norms and are equivalent. (d)Different choices of satisfying (17) will yield the equivalent norms in the definition of .

Let and . The Morrey space , which can be realized as , admits two different Littlewood-Paley characterizations. In [1], Mazzucato established
Meanwhile, combining this equivalence with what we proved in [15], we can say that
Thus, (23) is closer to Definition 3 than (22). Also, (23) seems to have stemmed from the famous technique due to Uchiyama [22]. We take advantage of equivalence of Theorem 4 in the proof of Theorem 6.

To establish Theorem 4, we will need an auxiliary vector-valued estimate of the Hardy-Littlewood maximal operator . Define the Hardy-Littlewood maximal operator by
The following proposition is proven in our earlier paper [12]. This is an extension of [23] to .

Proposition 5. Let , and . Assume in addition that . Then we have
for some independent of , where we modify (25) obviously when .

Chiarenza and Frasca obtained the boundedness of Hardy-Littlewood maximal operators on global Morrey spaces in [24]. In [11] Burenkov and Guliyev considered local Morrey-type spaces, where they showed that maximal operators are bounded [25–27].

With Theorem 4 and Proposition 5 in mind, we investigate the boundedness property of again as we announced in the beginning. More precisely, we shall provide an alternative proof of the following theorems, which were proven earlier in [10, 12].

Theorem 6 (see [10, 12]). Suppose that the parameters , and satisfy
Assume in addition that
Then is a bounded operator from to .

Theorem 7 (see [10, 12]). Suppose that the parameters , and satisfy (26) and
with . Then is a bounded linear operator from to .

We can also consider Campanato spaces and Lipschitz spaces in this framework. First, let be the set of all polynomials having a degree at most . For a cube , a locally integrable function over and a nonnegative integer , there exists a unique polynomial such that, for all ,
Denote this unique polynomial by . It follows immediately from the definition that if . We can characterize spaces and spaces (cf. [28]).

Definition 8. Let and . Then, for , define that
where we use the obvious modification when . The spaces and are the sets modulo of all for which the quantities and are finite, respectively.

Note that, in particular,
with norm coincidence. For the definition of and , we refer to [12].

Definition 9. For , , and with , let be the set of all such that , where
where we use the obvious modification when .

Observe that the quotient space is a Banach space equipped with the norm .

Proposition 10. Let , , . Assume that and that . Then
with equivalent norms.

Now we define a function space by way of difference. For , an integer and a function , we define
inductively.

Definition 11. For , and with , let be the set of all continuous functions such that , where
Then is a Banach space equipped with the norm
and also the quotient space is a Banach space equipped with the norm .

Definition 12. For and let be the set of all such that , where
In the present paper, we aim to show the following equivalence as well. Here, for , we let the largest integer such that .

Theorem 13. Let and . Assume that the integer satisfies . (1)If is a continuous function such that the quantity is finite, then and the inequality holds. (2)If , then can be represented by a continuous function and the inequality holds.

Remark 14. It is absolutely necessary to assume that is a continuous function in Theorem 13, when . We remark that there exists a discontinuous function such that for all . See [29, Proposition A1] for such an example constructed algebraically.

Now we explain notations and we describe its organization of the present paper. We use the following notations for the inequalities. First, we use standard notation for inequalities. For example, in the present paper a chain of inequalities of the form
appears in (110) below. The inequality (38) means that there exist such that
If the implicit constants in or do depend on some important parameters , then we write or . We shall prove Theorem 4 in Section 3. We prove Theorem 6 in Section 4. Theorem 7 is covered in Section 5. We prove Theorem 13 in Section 6. Finally in Section 7 we present another application of Theorem 4 by showing that the Fourier multiplier is bounded on .

2. Preliminaries

In the present paper, we frequently use the following fundamental inequalities.

Lemma 15 (see [30, page 466]). Let , and satisfy
Suppose that such that
for some . Assume in addition that is a measurable function such that
and that
for some . Then we have

To formulate the next lemma, we recall the definition of with . A measurable function , which takes values in almost everywhere, is said to be an weight or belongs to the class , if
For all , it is easy to see that and that .

Lemma 16 (see [31]). Let and satisfy (17). Then, for , we have
for all .

We also need a piece of information on dilation of the space .

Lemma 17. Let , and . Let and . Then

It is just a matter of handling the left-hand side carefully. But, for the sake of convenience, we supply the proof.

Proof. From the definition of , we deduce
proving the lemma.

In the course of the proof of Theorem 6, we need another piece of information on the space . Let , and . Define as the set of all such that , where

The next lemma concerns the norm of the translation operator.

Lemma 18. Let , and . Then

Proof. Let and be fixed and consider
Then we have
and . Note that
since . This implies that
If we consider the supremum over and such that , then we obtain the desired result.

Lemma 19. Let . Let , and . Then
for all .

Proof. We have
from Lemma 18. Hence, by the triangle inequality, we obtain that
Meanwhile, since , for , we have, on ,
Assuming that , we have
Putting (57)–(59) together, we obtain the desired result.

Lemma 20. Suppose that the parameters , and satisfy
Let . Then

Proof. Note that for . Recall that has a scaling invariance, as we have verified in Lemma 17. So, to prove (61), we can assume that . In this case, (61) is Lemma 19 itself.

We need the following sequence of functions.

Lemma 21. There exists a sequence of functions such that, for any fixed constant ,
provided .

In the present paper the sequence above is called a Rademacher sequence.

3. Littlewood-Paley Characterization of

It follows from the definition of the norm that
for any constant function . Note that this implies that is a subset of .

Let . Then, since , we can use the Hölder inequality and we have
for . Assuming that , the sequence is convergent by the Cauchy test. Thus, we can consider the mapping
Let us check that the range of is in .

Let and be fixed. Let be the largest integer such that . Then observe that . We decompose
We consider the -norm over and multiply .

As for the first term, directly from the definition of , we obtain that
Also, a geometric observation shows that the second term can be estimated similarly. Since , we obtain that
By using (64) and , we can handle the third term:
In view of the way in which we chose , we obtain that
In summary,
It remains to estimate the fourth term. We employ the following estimate:
Since , (72) is summable and we obtain that
where the implicit constant in (73) is independent of .

It follows from (67), (68), (71), and (73) that sends to boundedly;
for all .

Meanwhile, it follows from the norm that
for all . In view of (63), is a surjection:

Finally observe that, for , if and only if is a constant function, that is, . Namely,
Thus, from (75), (76) and (77), we conclude (33).

For and , let us define the function space of uniformly functions by
where, for , we write , and the norm is given by
Then from the definition of the norms (10) and (78), the following chain of continuous inclusion holds. For ,
Thus, (a) follows.

Remark 22. The space is a special case of amalgams investigated in [32].

Let be a fixed function satisfying (17). It follows from (17) that there exist and such that
Let and fix a dyadic cube such that . We are going to show that
converges in the topology of , and that
converges in the topology of .

The presice meaning of (82) and (83) is that as follows: (82) means
as and (83) means that
as . Once (82) and (83) are proven, we will have
converges in the topology of and that is a polynomial. Hence it follows that . Remark also that the convergence in of the sum defining is a generality. So let us prove (82) and (83).

Let us begin with proving (82). To this end we take a dyadic cube containing . Since we are assuming (81), we deduce that
Since , we have
from (19) and (20). By the Hölder inequality and the fact that , we have
for all . By decomposing the last integral dyadically, we obtain that
Observe from Definition 3 that
for all . Thus, we obtain
Hence, from (92) and the fact that , we have
which proves the embedding into . To prove that the sum defining converges in the topology of , we first fix a cube containing and observe
from (91) and the fact that . Meanwhile, again by (92), we have
for all cubes containing . As a result, we obtain that
from the fact that . It follows from (94) and (96) that the sum defining converges in .

Now let us prove (83). First, choose . By virtue of Lemma 16 and the fact that has a bound independent of [33], we obtain that
Here for the last inequality, we employed a geometric observation of the support of . Therefore, of the last inequality depends only on .

Since , we have
Recall that we are assuming that . So (98) is summable and we have